Among 5 people, what is the probability that exactly 2 of them are born in the same month.
For "at least $2$ in the same month", my answer would be
$$1 – \frac{12\cdot 11\cdot 10\cdot 9\cdot 8}{12^5}$$
the complement of probability that all months are distinct. How to deal with "exactly 2"?
(Assuming the birthdays are independent, and equally distributed among the months.)
Best Answer
It is tacitly assumed that there are $12^5$ equiprobable cases. We now have to count the favorable cases. There are ${5\choose2}=10$ ways to select the two people having birthday in the same month, and $12$ ways to select that month. There are three people left, to which we can assign a month each in $11\cdot10\cdot 9$ ways. The probability in question therefore comes to $$p={10\cdot 12\cdot 11\cdot 10\cdot 9\over 12^5}={275\over576}\doteq0.47743\ .$$